Question 1206200: B. Do the following.
1. Find the variance and standard deviation of the probability distribution of the X which can take only the values 1, 2, and 3, given that P(1)=\frac{10}{33}, P(2)=\frac{1}{3}, and P(3)=\frac{12}{33}.
2. Find the variance and standard deviation of the probability distribution of the variable X which can take only the values 3, 5, and 7, given that P(3)=\frac{7}{30}, P(5)=\frac{1}{3}, and P(7)=\frac{13}{30}.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
I'll focus on problem 1 only.
More narrowly, I'll focus on the variance only of this problem.
Many (online) calculators will quickly compute the variance of a discrete probability distribution. There's not much to explain other than "type in the numbers, push the right buttons, and out comes the answer".
But I'll show what is going on underneath the hood (i.e. how the calculator is computing the variance).
This is what the table looks like to compute the variance. The approximate variance is shown in red
| X | P(X) | X*P(X) | (X-mu)^2*P(X) | | 1 | 0.3030303 | 0.3030303 | 0.34087429 | | 2 | 0.33333333 | 0.66666666 | 0.00122436 | | 3 | 0.36363636 | 1.09090908 | 0.32089491 | | Sum | 2.06060604 | 0.66299356 |
mu = mean = expected value = E[X]
mu = sum of the X*P(X) values
mu = 2.06060604 approximately
The table was generated using LibreOffice spreadsheet (free spreadsheet software).
If you have Excel then use that instead. Another alternative would be GoogleSheets.
There are many alternatives to choose from. Feel free to search out your favorite.
The labels X*P(X) and (X-mu)^2*P(X) at the top of each column explains what is going on to each. If you're confused about this notation then please let me know.
Unfortunately the table may have some slight rounding error.
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Take notice that I used the approximate decimal form of each fraction (eg: 10/33 = 0.3030303).
The reasoning should be fairly obvious once we attempt to find the variance in terms of exact fractions only. Refer to the table below.
| X | P(X) | X*P(X) | (X-mu)^2*P(X) | | 1 | 10/33 | 10/33 | 12250/35937 | | 2 | 1/3 | 2/3 | 4/3267 | | 3 | 12/33 | 12/11 | 3844/11979 | | Sum | 68/33 | 722/1089 |
variance = 722/1089 = 0.66299356 approximately.
The fractions are very large and very messy.
The table was generated using a custom Python script tailored to find the variance in terms of exact fractions.
I couldn't find an online solver that would do it in terms of exact fractions, but one may be out there.
If your teacher is realistic, then s/he will hopefully accept the approximate version instead of the exact fraction form.
Sometimes fractions are too much of a pain to bother with.
Tip: Once you know the variance, apply the square root to find the standard deviation.
For example, if the variance is 49, then the standard deviation is 7
For more practice, refer to part (c) in this question
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1197443.html
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